Risk-sensitive Consumption and Savings under Rational Inattention∗ Yulei Luo†
Eric R. Young‡
University of Hong Kong
University of Virginia
June 20, 2009
Abstract This paper studies the consumption-savings behavior of households who have risk-sensitive preferences and suffer from limited information-processing capacity (rational inattention or RI). We first solve the model explicitly and show that RI increases precautionary savings by interacting with income uncertainty and risk-sensitivity. Given the closed-form solutions, we find that the RI model displays a wide range of observational equivalence properties, implying that consumption and savings data cannot distinguish between risk-sensitivity, robustness, or the discount factor, in any combination. We then show that the welfare costs from RI are larger for risk-sensitive households than any other observationally-equivalent settings. Thus, a potential method for testing risk-sensitivity vs. robustness is to ask agents how much they would pay to receive better information about the state of the world.
JEL Classification Numbers: C61, D81, E21. Keywords: Rational Inattention, Consumption, Precautionary Savings, Risk-sensitive Control, Robust Control. ∗
We thank Olivier Blanchard (The Editor), Thomas Muench, Jonathan Parker, Ricardo Reis, Chris Sims, Mirko Wiederholt, Chi-Wa Yuen, and an anonymous referee, as well as seminar and conference participants at the 2008 North American Summer Meetings of the Econometric Society, the 2008 Meeting of the Society for Economic Dynamics, the 2008 Midwest Macroeconomic Meetings, SUNY-Stony Brook, University of Hawaii at Manoa, the University of Hong Kong, and the Chinese University of Hong Kong for helpful comments and suggestions. Luo thanks the General Research Fund (GRF, formerly CERG) in Hong Kong for financial support. Young thanks the Bankard Fund for Political Economy at Virginia for financial support. All remaining errors are our responsibility. † Corresponding author. School of Economics and Finance, University of Hong Kong, Hong Kong. E-mail:
[email protected]. ‡ Department of Economics, University of Virginia, Charlottesville, VA 22904. E-mail:
[email protected].
1. Introduction Sims (1998) first argued that rational inattention (RI) is a plausible method for introducing sluggishness, randomness, and delay into economic models. In his formulation agents have finite Shannon channel capacity, limiting their ability to process signals about the true state of the world. As a result, an impulse to the economy induces only gradual responses by individuals, as their limited capacity requires many periods to discover just how much the state has moved. Sims (2003) and Luo (2008) use this model to explore anomalies in the consumption literature, particularly the wellknown ’excess sensitivity’ and ’excess smoothness’ puzzles, employing a linear-quadratic version of the standard permanent income model (as in Hall 1978). In that model RI is equivalent to confronting the household with a signal extraction problem regarding the value of permanent income (as in Friedman 1957) but permitting the agents to choose the distribution of the noise terms, subject to their limited capacity. With normal innovations to income the optimal distribution of noise is also normal, leading to a simple and analytically tractable problem. Of course, the linear-quadratic model has some undesirable features as well. One objection that is particularly relevant for this paper is that it satisfies the certainty equivalence property, ruling out any response of saving to uncertainty (that is, precautionary behavior). Given that the important component of RI is the introduction of endogenous uncertainty into the household problem, it is not particularly desirable to use a model in which this uncertainty cannot manifest itself in decision rules. Fully nonlinear versions of the RI problem are solved in Sims (2005, 2006), Lewis (2006), Batchuluun, Luo, and Young (2008), and Tutino (2008); these papers show that the precautionary aspect of RI is important when channel capacity is small (which it must be to produce any interesting results). But the models solved in those papers have either very short horizons or extremely simple setups due to numerical obstacles – the state of the world is the distribution of perceived states and this distribution is not well-behaved (it is not generally a member of a known class of distributions and tends to have ’holes,’ making it difficult to characterize with a small number of parameters). It is important to find a class of models that can produce precautionary behavior while maintaining tractability in the RI setup, if the properties of finite channel capacity are going to be thoroughly explored. One class of models that satisfies these desiderata is the risk-sensitive permanent income model
1
from Hansen, Sargent, and Tallarini (1999) (henceforth, HST).1 In this model agents effectively compute expectations through a distorted lens, increasing their effective risk aversion by overweighting negative outcomes.
The resulting decision rules depend explicitly on the variance of the shocks,
producing precautionary savings, but the value functions are still quadratic functions of the states, leading again to the optimality of Gaussian noise in the RI model. Thus, we preserve the tractability of the LQ PIH model without being forced to accept certainty equivalence; for future reference we denote the model with risk sensitivity and rational inattention as the RS-RI model.2 Formally, one can view risk-sensitive agents as ones who have non-state-separable preferences, as in Epstein and Zin (1989), but with a restricted value for the intertemporal elasticity of substitution (see Tallarini 2000). A second class of models that produces precautionary savings but remains within the class of LQ-Gaussian models is the robust control model of Hansen and Sargent (1995). In robust control problems, agents are concerned about the possibility that their model is misspecified in a manner that is difficult to detect statistically; as a result, they choose their decisions as if the subjective distribution over shocks was chosen by a malevolent nature in order to minimize their expected utility (that is, the solution to a robust decision-maker’s problem is the equilibrium of a max-min game between the decision-maker and nature). Hansen and Sargent (2007) present an observational equivalence result for RS and Robust models for consumption and savings decisions: any consumption path that could be generated by a model featuring risk sensitivity can also be generated by a model with robustness. Thus, introducing RI into the Robust model – which we denote the RB-RI model – is again straightforward, since the model retains the optimality of Gaussian noise. Our purpose in this paper is to study the properties of RS-RI and RB-RI models. In particular, we are interested in two main questions. The first question involves the question of observational equivalence – do RS-RI and RB-RI models possess combinations of parameters such that their implied consumption-savings rules are the same? And can those decision rules also be generated by a standard RI model? To preview the results, which we discuss in more detail below, the answer is yes – any member of the class of LQ-Gaussian models with information-processing capacity can produce 1
Possibly the first contribution to this literature is van der Ploeg (1993). Hansen and Sargent (2004) demonstrate that the risk-sensitive model displays a form of certainty equivalence, but that it generates precautionary savings anyway. 2 Of course, quadratic utility carries with it other objectionable properties, such as increasing relative risk aversion.
2
the same consumption function with appropriate adjustments in some combination of the discount factor, the degree of risk sensitivity, or the preference for robustness. The second question is how the costs of finite information-processing capacity differ within the class of observationally-equivalent models; that is, do observationally-equivalent agents suffer more or less from RI depending on which model generates their behavior? As a preview, we find here that the costs are not equal – RS-RI models produce larger welfare losses from RI than do RB-RI and RI models. Thus, we identify a potential channel through which risk sensitivity and robustness can be disentangled – simply ask agents what they would pay to relax their Shannon channel constraint.3 Our first step in studying the various RI models is to perform a reduction of the state space. Multivariate versions of the RI model require a constraint we term ’no subsidization.’ With more than one state variable, agents can allocate their attention differently across these variables and thus reduce their uncertainty at different rates; RI requires that the uncertainty regarding one variable cannot be increased in order to reduce uncertainty regarding another, as this would permit reductions that exceed the channel capacity. This constraint is nonlinear when the dimension is larger than one, even in the LQ model, so multivariate versions of the RI model are not any more tractable than the nonlinear models mentioned above. We show that a PIH model with a general labor income process can be reduced to a model with a unique state variable – permanent income – that has iid innovations, enabling us to study a model with an empirically-reasonable income process within the LQ setup. We then present the basic risk-sensitive (RS) model and derive some results regarding the behavior of consumption and saving. We show explicitly that both precautionary savings and the marginal propensity to consume out of permanent income are increasing in the parameter that measures risk sensitivity. From here we proceed by introducing finite Shannon channel capacity; RI implies that behavior is as if the agent were observing a noisy signal of current period’s true permanent income, so the agent uses the Kalman filter to update the perceived state. Our RS-RI model predicts that the precautionary saving increment is determined by the interaction of three factors: labor income uncertainty, enhanced risk aversion due to RS, and incomplete information due to finite channel capacity (RI). Specifically, we show that it increases with the degrees of income uncertainty, RS, and 3
Of course, figuring out how to conduct this experiment may be considerably more difficult.
3
RI.4 Since the evolution of the true state is affected by precautionary motives, the evolution of the perceived state is as well. Thus, RS and RI interact – while the household uses the same Kalman gain parameter independent of the degree of risk sensitivity, the mean value of the signal received increases with RS so the perceived state evolves differently. Because we are able to provide closed-form solutions to the RS-RI model we can provide two general results. First, introducing RI into the RS model improves the predictions about the aggregate joint dynamics of consumption and income; this result is obtained because the basic RS model actually reduces the ability of the model to match the sensitivity of consumption to income changes.5 Second, we show that observational equivalence holds for quantities – we can find more than one combination of discount factors and risk-sensitivity that imply the same savings and consumption behavior, whether RI is present or not. The intuition is the same as that in HST (1999) – an agent with RS looks just like a standard but more patient agent. We then turn to the RB-RI model.6 Here again we derive closed-form solutions and generate an observational equivalence result, which now extends in multiple directions. That is, we show that the RB-RI model produces the same consumption-savings decisions as the RS-RI model and, therefore, the RE and RI models. The intuition remains the same as above – with a preference for robust decision-making the agent chooses to save more than before, appearing more patient when viewed through the lens of a standard RE or RI model. Econometrically, these observational equivalence properties imply that it is impossible to test for the presence of RI, RS, or robustness using this model and consumption-savings data alone. Given the observational equivalence result, we ask whether anything fundamentally distinguishes the RS-RI model from the RI and RB-RI models. We find a key difference when we measure the welfare costs of RI – that is, how much would agents pay an agency to reveal the true state of the world, given that their own capacity is fixed exogenously? Equivalently, one can think of this question as asking what an agent would pay to increase their information-processing capacity to ∞. Luo (2008) studied the costs of RI in the standard model and concluded that they are trivial. 4
In a continuous-time setting, Wang (2004) shows that (exogenously) partial observability of individual components of labor income generates additional precautionary saving due to estimation risk. 5 This fact explains the use of habit formation and preference shocks in RS models (such as Hansen, Sargent, and Tallarini 1999). 6 Hansen, Sargent, and Wang (2002) consider a robust LQ permanent income model with filtering, and examine equilibrium market prices of risk. They model imperfect information by assuming that the consumer cannot observe the components in the aggregate endowment.
4
Generalizing his result to the RS-RI and RB-RI models, we find that the costs may not be trivial; in particular, agents in the RS-RI model can suffer very large losses from RI. In fact, we can prove that the RS-RI model implies that the welfare losses from RI are necessarily larger than those in RB-RI, and that this gap is increasing as channel capacity converges to 0. One key difference is that agents in the RS-RI model change their consumption behavior as their capacity changes, whereas RB-RI agents do not. The second key aspect is related to the asset pricing results in HST (1999) – RS-RI agents are more sensitive to the fluctuations in consumption and would pay more to remove them (mirroring the fact that the RS-RI model produces better asset pricing). The remainder of the paper is organized as follows. Section 2 presents three PIH models with general income processes and discusses some results regarding optimal consumption-savings decisions and the joint dynamics of aggregate consumption and income (these models are the standard PIH model, the RS model, and the RB model). Section 3 introduces RI into both the RS and RB models and the behavior of consumption and savings; we provide closed-form solutions for all models here. Section 4 presents our battery of observational equivalence results. Section 5 then discusses the welfare costs of RI. Section 6 concludes with a discussion of some potential directions that the literature could proceed.
2. Three Rational Expectations Permanent Income Models In this section, we present three benchmark models. In the first model, we show that within the LQG framework, the multivariate permanent income model with general income processes can be reduced to the univariate model with iid innovations to permanent income that can be solved in closed-form solution. We then present the second model in which risk-sensitivity is introduced into the univariate PIH model in Section 2.1. We finally introduce the third model in which the preference for robustness is incorporated into the first standard RE PIH model. 2.1. The Standard Permanent Income Model In the standard permanent income model households solve the dynamic consumption-savings problem
max E0
{ct }∞ t=0
"∞ X t=0
5
t
#
β u(ct )
(2.1)
subject to wt+1 = R(wt − ct ) + yt+1 ,
(2.2)
where u(ct ) = − 12 (ct − c)2 is the period utility function, c > 0 is the bliss point, equation (2.2) is the flow budget constraint, ct is consumption, wt is cash-on-hand (or market resources), yt is a general income process with Gaussian white noise innovations, β is the discount factor, and R is the constant gross interest rate at which the consumer can borrow and lend freely.7 For the rest of the paper we will restrict attention to points where ct < c, so that utility is increasing and concave. This specification follows that in Hall (1978) and Flavin (1981) and implies that optimal consumption is determined by permanent income:
ct =
1 1− βR2
1 st − R−1
where st = wt +
∞ X
1 1− βR
c,
R−j Et [yt+j ]
(2.3)
(2.4)
j=1
is the expected present value of lifetime resources, consisting of financial wealth plus human wealth. If st is defined as a new state variable, we can reformulate the above PIH model as
v(s0 ) = max E0 {ct }
"∞ X
#
β t u(ct )
t=0
(2.5)
subject to st+1 = R(st − ct ) + ζ t+1 ;
(2.6)
a formal proof is contained in Luo (2008). The stochastic term ζ t+1 is equal to ζ t+1 =
∞ X
R−j+(t+1) (Et+1 − Et ) [yj ] ;
(2.7)
j=t+1
7 As in Hansen and Sargent (2007) or HST (1999), we can also interpret this problem as applying to a linear production technology that generates consumption and capital,
ct + kt = Rk kt−1 + yt , where Rk is the net physical return on capital and investment is given by it = kt −kt−1 . If the allocation is decentralized the return coincides with the risk-free interest rate R.
6
that is, ζ t+1 the time (t + 1) innovation to permanent income. v(s0 ) is the consumer’s value function under RE. By defining a new state variable st , the original multivariate optimization problem has been reduced to a univariate problem; under quadratic utility this model leads to the well-known random walk result of Hall (1978) when βR = 1:
∆ct+1 =
R−1 ζ ; R t+1
(2.8)
in this case, the change in consumption depends neither on the past history of labor income nor on anticipated changes in labor income. We also point out the well-known result that the standard PIH model with quadratic utility implies the certainty equivalence property holds; thus, uncertainty has no impact on optimal consumption, so that there is no precautionary saving. We assume for the remainder of this section that βR = 1, since this setting is the only one that implies stationary consumption. Let kt = wt − ct denote end of period financial wealth. Then it = kt − kt−1 is a measure of saving (or investment). Substituting this expression into (2.2) and using (2.6) and (2.8) we obtain ∆kt+1 = 0; thus, there is no wealth accumulation in response to changes in permanent income, as all such changes are allocated to consumption. For simplicity, following Pischke (1995) we suppose that income yt can be expressed as the sum of aggregate permanent and idiosyncratic transitory components: p t + yt+1 , yt+1 = yt+1
(2.9)
where the superscripts p and t denote permanent and transitory, respectively. Each of these compop nents follows its own stochastic process; yt+1 follows a random walk
p yt+1 = ytp + εt+1 ,
(2.10)
t yt+1 = y + ǫt+1 ,
(2.11)
t and yt+1 follows an iid process
where εt+1 and ǫt+1 are white noises with mean 0 and variance ω 2ε and ω 2ǫ respectively. Substituting
7
these income processes into (2.8) gives
∆ct+1 = εt+1 +
because the innovation to permanent income ζ t+1 =
R−1 ǫt+1 R R R−1 εt+1
(2.12)
+ ǫt+1 . The different load factors on
the two shocks reflects the standard consumption-smoothing motives: consume all of a permanent increase in income (εt+1 ) but only the annuity value of a transitory one (ǫt+1 ). We now want to derive some properties of aggregate consumption. Suppose that there are a continuum of consumers in the model economy. Given the expression of the change in individual consumption, (2.12), aggregating across all consumers yields the change in aggregate consumption as ∆Ct+1 = εt+1 ,
(2.13)
so that aggregate consumption Ct+1 is unpredictable using past information and the smoothness ratio of aggregate consumption to income is 1.8
In other words, the predictions of the standard
PIH model for the joint behavior of aggregate consumption and income is not consistent with the empirical evidence; in the US data aggregate consumption growth is much smoother than income and is sensitive to past information.9 These two anomalies have been termed the excess smoothness and excess sensitivity puzzles in the literature.10 Furthermore, as documented in Reis (2006), the impulse response of aggregate consumption to aggregate income takes a hump-shaped form, which means that aggregate consumption reacts to income shocks gradually. The standard RE PIH model cannot capture this feature in the US data. 2.2. The Risk-sensitive Permanent Income Model Risk-sensitivity (RS) was first introduced into the LQ-Gaussian framework by Jacobson (1973) and extended by Whittle (1981, 1990). Exploiting the recursive utility framework of Epstein and Zin (1989), Hansen and Sargent (1995) introduce discounting into the RS specification and show that the resulting decision rules are time-invariant. HST (1999) use this model to study a risk-sensitive version of the PIH model and show how RS alters the choices of consumption, investment, and asset 8
The idiosyncratic shock, ǫ, would be canceled out after aggregating over all consumers. In the US data, the smoothness ratio is around 0.5. 10 See Deaton (1992) for a recent review. 9
8
prices; Tallarini (2000) addresses many of the same issues within the stochastic growth model. We use an RE version of risk-sensitive control based on Kreps and Porteus (1978) preferences with an exponential certainty equivalence function. The household problem can be written recursively as 1 2 v (st ) = max − (ct − c) + βRt [v (st+1 )] , ct 2
(2.14)
subject to (2.6). The distorted expectation operator Rt is defined by
Rt [v (st+1 )] = −
1 log Et [exp (−αv (st+1 ))] , α
(2.15)
where α > 0 measures higher risk aversion vis a vis the von Neumann-Morgenstern specification. Solving this optimization problem yields the following proposition: Proposition 1. The value function for the RS model is R βR2 − 1 1 βR2 − 1 2 v (st ) = − s + cst − 2 βR2 − αω 2ζ t βR2 − αω 2ζ (R − 1) ! 2 2 2 R βR − 1 β 1 βR − 1 1 c2 − log 1 − αω 2ζ , 2 2 (R − 1) βR2 − αω 2 1 − β 2α βR2 − αω 2ζ
(2.16)
ζ
the consumption function is 2 + (1 − R) αω 2 c R − βR ζ −1 st + ct = . 2 2 βR − αω ζ βR2 − αω 2ζ (R − 1) βR2
(2.17)
Proof. See Appendix 7.1. Equation (2.17) shows that the parameter controlling risk-sensitivity, α, affects both the marginal 2
βR −1 propensity to consume (MPC) out of perceived permanent income, βR 2 −αω 2 , and the precautionary ζ 2 (R−βR2 +(1−R)αω 2ζ )c R 2 2 2 11 savings increment, − βR2 −αω2 (R−1) , where ω ζ = R−1 ω ε + ω ǫ . The larger the value of α ( ζ) the larger the MPC and the larger the precautionary saving increment (we impose the restriction 11
Given the definition st in (2.6), the MPC out of permanent income equals the MPC out of financial wealth wt P of −j and human wealth ∞ R Et [yt+j ]. j=1
9
βR2 > 1 > αω 2ζ so that precautionary savings is positive), since ∂
βR2 −1 βR2 −αω 2ζ
∂α
(R−βR2 +(1−R)αω2 )c ∂ − βR2 −αω2 (R−1)ζ ( ζ) > 0 and ∂α
> 0.
As its name suggests, risk sensitivity makes agents more concerned about outcomes that are unfavorable; one way to minimize the probability of bad outcomes (here, outcomes where the marginal utility of consumption becomes very large) is to save a bit more, reducing consumption relative to the certainty equivalent setup where α = 0. In HST (1999) and Hansen and Sargent (2007), they interpret the RS preference in terms of a concern about model uncertainty (robustness or RB) and argue that RS introduces precautionary savings because RS consumers want to protect themselves against model specification errors.12 It is worth noting that this type of precautionary saving does not depend on the convexity of the marginal utility of consumption and occurs even when the period utility of consumers is quadratic as we have shown. In the long run agents will have more wealth as the extra precautionary savings generates more returns. Relative risk aversion over wealth gambles is given by rra (st ) = function of
c st .
1 R c −1 R−1 st
, which is an decreasing
Thus, α does not directly impact risk aversion in this model; what matters for risk
aversion is only how close current permanent income is to the bliss point. α increases st , however, and thus indirectly increases the risk aversion of the agent; the increasing relative risk aversion property of quadratic utility functions remains in the RS setting. We now present some basic results from the RS model, mainly to facilitate comparisons to the RI model of the next section. The change in consumption can be written as
∆ct+1 = −
βR2 − R − (1 − R) αω 2ζ βR2 − αω 2ζ
(ct − c) +
βR2 − 1 ζ . βR2 − αω 2ζ t+1
(2.18)
Equation (2.18) has two important implications for our purposes. Setting α > 0 implies that consumption growth becomes more volatile relative to income and becomes more sensitive to unanticipated changes in permanent income, since
βR2 −1 βR2
< 1. Furthermore, consumption growth is not
mean zero unless ct = c even when βR = 1; indeed, when βR = 1 consumption will drift upward 12
Later we show that although RS and RB are observationally equivalent in the sense that they cannot be distinguished by consumption, savings and investment data alone under RE or RI, they have different implications for welfare and stabilization policies. Hence, we should be careful when using one to interpret the other.
10
b < 1 such that the tendency to save for precautionary over time. Counteracting this drift requires βR
reasons is exactly counterbalanced by the impatience effect of increasing consumption; this result is the basis of the observational equivalence result in HST (1999). As in the last subsection, substituting the specified income processes, (2.9), (2.10), and (2.11), into (2.18) gives
∆ct+1
βR2 − 1 = βR2 − αω 2ζ
"
R εt+1 + ǫt+1 R−1
# + ǫt + (ρ1 − 1) , 1 − ρ1 L R R−1 εt
(2.19)
1−αω 2
ζ where ρ1 = R βR2 −αω 2 ∈ (0, 1) and L is the lag operator. Wealth accumulation (∆k) is given by ζ
∆kt+1 =
1 − αω 2ζ βR2
−1
!
∆ct+1 −
1 ∆yt+1 ; R−1
(2.20)
thus, risk-sensitive consumers adjust their optimal plan by consuming only a fraction of any increase in income. Assets are reduced when income growth is negative and increased income growth is positive. Aggregating (2.19) across all consumers gives the change in aggregate consumption:
∆Ct+1
R βR2 − 1 εt = εt+1 + (ρ1 − 1) . R − 1 βR2 − αω 2ζ 1 − ρ1 L
(2.21)
Figure 1 illustrates the response of aggregate consumption growth to an aggregate income shock εt+1 ; RS raises the sensitivity of consumption growth to unanticipated changes in aggregate income. Furthermore, the smoothness ratio of aggregate consumption to income is seen to be sd [∆Ct ] R βR2 − 1 µ= = sd [εt ] R − 1 βR2 − αω 2ζ
s
2 , 1 + ρ1
(2.22)
where “sd” denotes standard deviation. It is straightforward to prove the following proposition, which is the key point from this section. Proposition 2.
∂µ ∂α
> 0.
11
Proof. Clearly the first term is increasing in α. Then ω 2ζ (1 − R) ∂ρ1 = R 2 < 0. ∂α R − αω2ζ Thus, µ is increasing in α. RS increases the response of aggregate consumption growth to a change in income, just as it does for individual income. Thus, RS by itself worsens the standard RE PIH model’s prediction for the joint behavior of aggregate consumption and income growth by exacerbating the excess smoothness puzzle, and therefore needs to be combined with other assumptions to resolve the anomalies.13 2.3. The Robust Permanent Income Model Robust control emerged in the engineering literature in the 1970s, and was introduced into economics and further developed by Hansen, Sargent, and others. A simple version of robust optimal control considers such a question: How to make decisions when the agent does not know the probability model that generates the data? The agent with the preference for robustness considers a range of models, and makes decisions that maximize utility given the worst possible model. Following Hansen and Sargent (2007), the simple robust permanent income problem can be written as 1 v (st ) = max min − (c − ct )2 + β ϑν 2t + Et [v (st+1 )] ct νt 2
(2.23)
st+1 = R (st − ct ) + ζ t+1 + ω ζ ν t ,
(2.24)
subject to
where ν t distorts the mean of the innovation and ϑ > 0 controls how bad the error can be.14 Solving this robust control problem yields the following proposition:15 13 In HST (1999), habit persistence is incorporated into the RS version of the PIH model. Hansen and Sargent (2007) use habit persistence and/or adjustment costs. 14 Formally, this setup is a game between the decision-maker and a malevolent nature that chooses the distortion process ν t . ϑ ≥ 0 is a penalty parameter that restricts attention to a limited class of distortion processes; it can be mapped into an entropy condition that implies agents choose rules that are robust against processes which are close to the trusted one. 15 See Backus, Routledge, and Zin (2004) for a detailed derivation.
12
Proposition 3. The value function for the above RB model is βR2 − 1 R βR2 − 1 2 − cst s t R − 1 βR2 − ω2ζ / (2ϑ) 2βR2 − ω 2ζ /ϑ 2 2 2 R βR − 1 β βR − 1 2 c2 + − , 2 /ϑ ω ζ 2 2 2 1 − β 2βR − ω ζ 2 (R − 1) βR2 − ω ζ / (2ϑ)
v (st ) = −
(2.25)
and the consumption function is 2 / (2ϑ) + 1 − βR2 (R − 1) 1 − ω ζ −1 ct = st + . 2 2 βR − ω ζ / (2ϑ) (R − 1) βR2 − ω 2ζ / (2ϑ) βR2
(2.26)
The consumption function under RB, (2.26), shows that the preference for robustness, ϑ, affects both the MPC out of permanent income,
βR2 −1 , βR2 −ω 2ζ /(2ϑ)
and the precautionary savings increment,
) . The less the value of ϑ the larger the MPC and the larger the precautionary ( ) saving increment (we impose the restriction βR2 > 1 > ω 2ζ / (2ϑ) so that precautionary savings is −
(
(R−1) 1−ω 2ζ /(2ϑ) +1−βR2 (R−1) βR2 −ω 2ζ /(2ϑ)
positive). Furthermore, comparing (2.26) with (2.17), it is clear that when α = 1/ (2ϑ), the RS and RB models are observationally equivalent (OE) in the sense that they generate the same consumption and savings decisions, as well as the same joint dynamics of consumption and income. For this reason, HST (1999) and Hansen and Sargent (2007) interpret the RS preference in terms of a concern about model uncertainty (robustness or RB) and argue that RS introduces precautionary savings because RS consumers want to protect themselves against model specification errors. However, (2.16) and (2.25) clearly show that the two models under OE have different welfare implications, as βR2 − 1 1 − log 1 − αω 2ζ α βR2 − αω 2ζ
!
>
βR2 − 1 ω2 βR2 − ω 2ζ / (2ϑ) ζ
for α > 0. In other words, RS consumers are more risk averse than RB consumers, and thus are willing to pay more to avoid the uncertainty due to labor income. Furthermore, since stabilization policies are closely related to welfare costs due to uncertainty, the two models would have different implications for stabilization policies, and thus they cannot be viewed observationally equivalent in this aspect.
13
3. Permanent Income Models with Rational Inattention We now introduce rational inattention (RI) by assuming the consumer faces information processing constraints; as first suggested in Sims (1998), households have only finite Shannon channel capacity that they can apply to observations of the state of the world. As in Sims (2003), we use the concept of entropy from information theory to characterize the rate of information flow; the reduction in entropy is a natural measure of information flow, where entropy is defined as a measure of the uncertainty about a random variable.16 With finite channel capacity, the consumer will choose a signal that reduces the uncertainty of the state variable subject to limitations on the extent that entropy can be reduced. Formally, this idea is described by the information processing constraint
H (st+1 |It ) − H (st+1 |It+1 ) ≤ κ,
(3.1)
where κ is the consumer’s channel capacity that imposes an upper bound on the amount of information that can be transmitted via the channel, H (st+1 |It ) denotes the entropy of the state prior to observing the new signal at t + 1, and H (st+1 |It+1 ) denotes the entropy after observing the new signal.17 Given the LQG specification, Sims (2003, 2005) showed that (st |It ) is distributed N sbt , σ 2t ,
where sbt = Et [st ] and σ 2t = vart [st ] are the conditional mean and variance of the state variable st ,
respectively. In models with imperfect observations, the optimal decisions are determined by the perceived state, rather than the actual state. Computational difficulties arise in RI models because the perceived state is the distribution of the actual state variable conditional on the information set available at time t, It , and as we have noted already this object typically has high dimension.
Fortunately, given the LQG specification and the Gaussian distribution of st , the first two moments, sbt and σ 2t , are enough to characterize the perceived state. In addition, as we will show, the problem will simplify further because σ 2t converges to a constant in the stochastic steady state. 16
See Shannon (1948) and Cover and Thomas (1991) for details. We regard κ as a technological parameter. If the base for logarithms is 2, the unit used to measure information flow is a ‘bit’, if we use the natural logarithm e, the unit is a ‘nat’, and if we use a base 10 the unit is a ‘dit’ (also called a ‘ban’ or a ‘hartley’). 1 nat is equal to log 2 e = 1.433 bits. We will use the nat in this paper. Lewis (2006) studies a model where households can increase their channel capacity by reducing their leisure; we have not investigated whether that model can be mapped into the LQ-Gaussian framework. 17
14
The constraint (3.1) can be rewritten as log ψ 2t − log σ 2t+1 ≤ 2κ,
(3.2)
where σ 2t+1 = vart+1 [st+1 ] and ψ 2t = vart [st+1 ] = R2 σ 2t + vart ζ t+1 are the posterior and prior variance of the state variable, st+1 , respectively.18 As shown in Sims (2003), in any univariate case,
this information constraint completes the characterization of the optimization problem with RI and the model can be solved explicitly. Furthermore, with a finite capacity κ the optimizing consumer will choose a signal that reduces the conditional variance of st+1 by a limited amount. Hence, (3.2) must be binding for the optimizing consumer: log ψ 2t − log σ 2t+1 = 2κ.
(3.3)
It is straightforward to show that in the univariate case (3.3) has a steady state σ 2 . In that steady R 2 2 ω 2ζ 2 state, σ 2t = σ 2 = exp(2κ)−R ω ε + ω 2ǫ , and the consumer behaves 2 , where ω ζ = vart ζ t+1 = R−1
as if he is observing a noisy signal, s∗t+1 = st+1 + ξ t+1 , where ξ t+1 is the iid endogenous noise and its variance, ̺2t = vart ξ t+1 , is determined by the usual updating formula of the variance of a Gaussian distribution:
σ 2t+1 = ψ 2t − ψ 2t ψ 2t + ̺2t
−1
ψ 2t .
(3.4)
Thus, in the steady state we have
̺2t = ̺2 = vart ξ t+1 =
ω2ζ + R2 σ 2 σ 2
ω 2ζ + (R2 − 1)σ 2
.
For simplicity, we assume that initially the model economy is at the steady state, implying s0 |I0 ∼ N sb0 , σ 2 . To draw attention to the value of reducing the multivariate PIH model to a univariate one we
note that RI with multiple state variables requires a second constraint:
Ψt − Σt 0, 18
Note that here we use the fact that the entropy of a Gaussian random variable is equal to half of its logarithm variance plus some constant term.
15
which is to be read as ’the matrix defined as the difference between the posterior and the prior element-by-element must be positive semidefinite.’ We term this constraint a ’no subsidization’ condition, as it implies that an agent cannot reduce one variable’s entropy by more than κ by increasing the entropy of another. This constraint takes the form of restrictions on eigenvalues and thus is not linear except in the univariate case where it is unnecessary.19 As a result, the multivariate version of the RI PIH model does not preserve the optimality of Gaussian noise; we therefore exploit the state reduction presented above and use permanent income as the unique state variable. Sims (2003) and Luo (2008) introduce RI into Hall’s permanent income models, and show that RI can provide an endogenous propagation mechanism that disentangles the short-run and long-run responses of consumption to income shocks (see the dotted-dashed line in Figure 1). In particular, Luo (2008) introduces information-processing constraints, (3.1), into the standard PI model proposed in Section 2.1 and solves for the following expression for consumption growth:
∆ct = θ
R−1 R
ζt 1 − (1 − θ)RL
+ ξt −
θRξ t−1 1 − (1 − θ)RL
;
(3.5)
this MA(∞) expression clearly shows that with finite capacity, consumption reacts to the income shock gradually and with delay. Comparing with the predictions of the RS-PIH model (see the solid line in Figure 1), it is clear that RS (or RB) and RI have distinct effects on the responses of consumption to income shocks: RS (or RB) just increases the sensitivity of consumption to the unanticipated income shock, whereas RI dampens the consumption reaction and thus generates a hump-shaped response to the shock.20
Kasa (2006) shows in a continuous-time filtering model
that RI and RB are observationally equivalent, in the sense that a higher filter gain can either be interpreted as a stronger preference for robustness or an increased information-processing capacity; however, as shown above, RI and RB (or RS) are totally different, in the sense that they generate different dynamics of consumption and savings. Luo (2008) contains a complete discussion of the basic linear-quadratic-Gaussian RI model, so we proceed by introducing two models that generalize the RI model to permit precautionary effects. The first, denoted RS-RI, introduces the risk sensitivity motive that we discussed in the previous 19 It is also likely that some variables would generate corner solutions where their entropy is not reduced at all, further exacerbating the nonlinearity of the constraint. 20 Note that RS and RB generate the same consumption-savings rules in the permanent income model.
16
section. The second, denoted RB-RI, assumes that agents have a demand for robust decision rules (those that are good against a wide range of misspecifications in the model they use for forecasting income movements). We lay out each model and solve it in this section. 3.1. RS-RI Model The household problem is characterized by the Bellman equation 1 2 vb (b st ) = max − (ct − c) + βRt [b v (b st+1 )] ct 2
(3.6)
subject to the budget constraint, (2.6), and the Kalman filter equation
st − ct ) + θ st+1 + ξ t+1 . sbt+1 = (1 − θ) R (b
(3.7)
The distorted expectation operator is now given by
Rt [b v (b st+1 )] = −
1 log Et [exp (−αb v (b st+1 ))] , α
(3.8)
s0 | I 0 ∼ N sb0 , σ 2 , sbt = Et [st ] is the perceived state variable, θ is the optimal weight on the new
observation of the state, and ξ t+1 is the endogenous noise. The optimal choice of the weight θ is given by θ (κ) =
exp(2κ)−1 exp(2κ)
∈ [0, 1]; therefore, we choose to parameterize RI using θ directly. Furthermore,
limκ→∞ θ (κ) = 1, so that we nest the standard RS specification as a special case of our model. To make economic sense in subsequent results we impose the condition θ >
R2 −1 R2
>
R−1 R ,
which implies
that agents must be able to process some minimum amount of information. This condition is similar to one imposed in Luo and Young (2008) that guarantees an agent with higher capacity optimally chooses a noise process with lower variance; we regard this requirement to be intuitively reasonable.21 The following proposition summarizes the solution to the RI-RS model. Proposition 1. Given finite channel capacity κ and the degree of risk-sensitivity α, the value func21
This assumption is not needed in fully nonlinear versions of the RI model, suggesting an important breakdown of the linear-quadratic setup as channel capacity gets very small.
17
tion of a risk-sensitive consumer under RI is R 1 − βR2 βR2 − 1 2 vb (b st ) = − sb + cb st − 2βR2 − 2αω 2η t βR2 − αω 2η (R − 1) " !# R2 βR2 − 1 β 1 βR2 − 1 1 2 2 log 1 − 2α ω , c − 2 (R − 1)2 βR2 − αω 2η 1 − β 2α 2βR2 − 2αω 2ζ η
(3.9)
and the consumption function is
1 − αω 2η (R − 1) + 1 − βR2 βR2 − 1 ct = sbt + c, βR2 − αω 2η βR2 − αω 2η (R − 1)
where
ω 2η = var η t+1 = and η t+1
(3.10)
θ ω2 1 − (1 − θ) R2 ζ
ζ t+1 θRξ t =θ + ξ t+1 − . 1 − (1 − θ)RL 1 − (1 − θ)RL
Proof. See Appendix 7.2. To interpret these expressions, we remind the reader that ξ t is the noise shock induced by RI and ζ t is the innovation to permanent income. η t+1 will turn out to be the stochastic component in the growth rate of consumption (see below); unlike the basic RS model, η t+1 contains a forecastable stochastic component due to the presence of the MA(∞) term. There are two special cases of interest that are nested in our specification: if α = 0 we obtain the same expressions as in Luo (2008) (the basic RI model), and if θ = 1 we obtain η t+1 = ζ t+1 (the basic RS model). Note that the permanent income shock ζ t and the noise shock ξ t have the same effect on sbt+1
(they both increase perceived permanent income by θ ≤ 1) but only ζ t affects st+1 (it increases by 1); because the variance of the noise shock will be decreasing in θ, agents with high processing capacity will end up with sbt tracking st quite closely. The consumption function shows that θ affects 2
βR −1 both the MPC out of the perceived state variable βR 2 −αω 2 and the precautionary savings increment η 2 2 (1−αωη )(R−1)+1−βR only through the variance of the noise distribution. With finite capacity ω 2η > ω 2ζ (βR2 −αω2η )(R−1) and it then follows that ω 2η is decreasing in θ.
Proposition 2.
∂ω 2η ∂θ
< 0.
18
Proof. By simple calculation we obtain 1 − R2 ω 2ζ ∂ω 2η = <0 ∂θ (1 − (1 − θ) R2 )2 because R > 1 and 1 − (1 − θ) R2 > 0. Thus, both the MPC out of the perceived state variable (the responsiveness of ct to sbt ) and
the precautionary savings increment (the intercept of the consumption profile) are decreasing in θ
for fixed α, meaning that agents with lower capacity save a larger fraction of any perceived income increase.
Figure 2 illustrates how the combination of (α, θ) affects the MPC out of financial and
human wealth (which are equal here) when R = 1.01. The relative risk aversion expression is now rra (b st ) =
1 R c −1 R−1 s bt
; the key difference is that risk aversion depends not on how far actual permanent
income is from the bliss point but rather how far perceived permanent income is from the bliss point. An RS-RI agent is more risk averse than a standard RS agent whenever sbt > st , which occurs when the agent receives negative shocks to ζ or positive shocks to ξ.
Since agents with low capacity are very concerned about the confluence of low permanent income and high consumption (meaning they believe their permanent income is high so they consume a lot and then their new signal indicates that in fact their permanent income was low), they take actions which reduce the probability of this bad event – they save more.22 The strength of the precautionary effect is positively related to the amount of uncertainty regarding the true level of permanent income, and this uncertainty increases as θ gets smaller. Figure 3 illustrates the effect of (α, θ) on the precautionary savings increment. In the absence of risk sensitivity (α = 0) the precautionary savings increment is zero independent of θ; if α > 0, however, precautionary savings is decreasing in θ, so that agents with lower capacity will consume less on average.23 Therefore, the precautionary savings increment in the RS-RI model is determined by the interaction of three factors: labor income uncertainty, enhanced risk aversion relative to the expected-utility setting (RS), and finite information-processing capacity (RI). Many empirical studies have estimated the importance of precautionary saving when people are exposed to future income uncertainty. For example, Carroll and Samwick (1998) estimate the precautionary savings for typical U.S. households to be in the 22
A more complete discussion of precautionary savings in an RI model can be found in Batchuluun, Luo, and Young (2008). 23 Note that agents with RI display precautionary behavior with respect to second-order increases in risk without requiring convex marginal utility, whereas RE households do not (see Eeckhoudt and Schlesinger 2008).
19
range of 20 − 50% of total savings, while Guiso, Jappelli, and Terlizzese (1992) use a survey of Italian households to estimate that the precautionary component explains only a small fraction of total saving. Hence, the result in Guiso, Jappelli, and Terlizzese (1992) casts doubt on the empirical relevance of precautionary saving generated by labor income uncertainty, but does not contradict the importance of the precautionary motive per se.
Besides exogenous income uncertainty, the
endogenous uncertainty due to RI generated from our RS-RI model may be a key determinant of precautionary savings. 3.1.1. Individual Dynamics The change in individual consumption in the RI-RS economy can be written as
∆ct+1
R − βR2 + (1 − R) αω 2η βR2 − 1 = (c − c)+θR t βR2 − αω2η βR2 − αω 2η
(1 − θ) ζ t − θξ t βR2 − 1 +θ ζ + ξ t+1 t+1 1 − (1 − θ) RL βR2 − αω 2η
so that ∞ ∞ ∞ R − βR2 + (1 − R) αω 2 X X X −1 η ∆ct = θ Γ ζ + Γ ξ − R Γ ξ + c (3.11) j j j t−j t−j t−1−j 2 2 2 2 βR − αωη βR − αωη βR2
j=0
where ρ1 =
R−αω 2η R βR2 −αω 2η
j=0
j=0
∈ (0, 1), ρ2 = (1 − θ) R ∈ (0, 1), and Γj =
P j−k k j−1 j−1−k k ρ ρ − ρ ρ 2 2 . 1 1 k=0 k=0
Pj
Thus, consumption growth is seen to be a weighted average of all past permanent income and noise shocks. Wealth accumulation under RI is
∆kt+1 =
1 − αω 2η 1 ∆ct+1 − ∆yt+1 + ∆ (st+1 − sbt+1 ) ; βR2 − 1 R−1
under RI the actual savings of risk-sensitive consumers responds to unperceived changes in the true state relative to the perceived one. Figure 1 also plots the responses of consumption to the aggregate permanent income shock εt . It clearly shows that RI has a similar qualitative effect on consumption in the standard RE model and the RS model – consumption reacts gradually to income shocks, with monotone adjustments to the corresponding asymptote – while quantitatively RI has larger impacts on consumption in the RS model because the deviation of the asymptote from that for the standard RE case is larger in this case. With a stronger preference for risk-sensitivity, the precautionary savings increment is larger 20
and thus an income shock that is initially undetected would have larger impacts on consumption during the adjustment process. In addition, we note that the immediate response of consumption to a shock to permanent income is given by
SR (εt ) = θ
βR2 − 1 . βR2 − αω 2η
(3.12)
This expression is increasing in α, implying that the more risk-sensitive the household the more consumption responds initially to changes in permanent income. It is also increasing in θ, implying that agents with lower capacity have smaller initial responses, so that RI and RS have opposing effects. In Figure 4 we plot the short-run responses of consumption growth as a function of θ for various values of α. With RI agents respond slowly, similar to their response if they faced convex adjustment costs for their wealth; one resulting implication is that the short-run and long-run responses of consumption are different. The long-run response is given by P T ∂( Ts=0 ∆ct+s ) βR2 − 1 X T −k k LR (εt ) = =θ ρ1 ρ2 . ∂εt βR2 − αω 2η
(3.13)
k=0
Note that when α = 0 and ρ1 = 1 and taking the limit as T → ∞ this expression reduces to the result reported in Luo (2008): LR (εt ) =
θ . 1 − ρ2
(3.14)
It is clear from the above two expressions that RS has two opposite impacts on the long-run responses of consumption to income shocks under RI. First, α increases the long-run response by a factor PT T −k k R 1 θ R−αω ; second, it reduces the long-run response by a factor lim ρ2 = 0 < 1−ρ T →∞ 2 k=0 ρ1 2
η
because |ρ1 | < 1. Furthermore, the volatility of individual consumption growth is
var [∆ct ] =
θ
R−1 R − αω 2η
2 X ∞
j=0
Γ2j ω 2ζ +
∞ X j=0
# j−1 j−2 X 2 X Γj − (1 + R) ρj−1−k ρk2 + ρj−1−k ρk2 ω 2ξ . 1 1
"
k=0
k=0
(3.15)
Intuitively, this expression should be decreasing in θ and increasing in α.
With lower θ, agents
have less channel capacity and therefore choose more volatile noise shocks, leading to changes in
21
consumption that are more sensitive to changes in perceived permanent income.24 3.1.2. Aggregate Dynamics Since the expression for the change in individual consumption (3.11) permits exact aggregation, we can obtain the change in aggregate consumption as ∞ ∞ ∞ X X X −1 i i , ∆Ct = θ Γj εt−j + Γj E ξ t−j − R Γj E ξ t−1−j βR2 − αω 2η βR2
j=0
j=0
(3.16)
j=0
where i denotes a particular individual and E i [·] is the population average. This expression shows that even if every consumer only faces the common shock ε, the RI economy still has heterogeneity since each consumer faces the idiosyncratic noise induced by finite channel capacity. As argued in Sims (2003), although the randomness in an individual’s response to aggregate shocks will be idiosyncratic because it arises from the individual’s information-processing constraint, there is likely a significant common component; provided that agents face similar needs for coding macroeconomic information, they will rely on common sources and generate aggregate noise shocks. Existing theory does not provide a way to determine the common component of the noise term; we can only state that the common term E i ξ it is between 0 and the part of the idiosyncratic error ξ t generated by the
aggregate income shock εt . Formally, assume that ξ t consists of two independent noises: ξ t = ξ t + ξ it , where ξ t = E i [ξ t ] and ξ it are the common and idiosyncratic components of the error generated by ζ t , var[ξ ] respectively. A parameter, λ = var[ξ t ] ∈ [0, 1], can be used to measure the common source of coded t
information on the aggregate component (or the relative importance of ξ t to ξ t ). In order to simplify expressions we first consider the case where all noises are idiosyncratic, that is, λ = 0 (so that individuals live on isolated islands and do not interact with each other directly or indirectly via conversation, imitation, newspapers, or other media); in this special case the change in aggregate consumption can be written as ∞ βR2 − 1 X ∆Ct = θ Γ ε . j t−j βR2 − αω 2η
(3.17)
j=0
24
The fact that the noise shocks are necessarily more volatile with lower channel capacity is a consequence of normality. Otherwise, the distribution of noise shocks would merely have more entropy, which might not necessarily translate into higher variance.
22
Therefore, the volatility of aggregate consumption relative to income can be written as
µ=
sd [∆Ct ] = sd [εt ]
v uX u∞ 2 −1 t θ Γj , βR2 − αω 2η βR2
(3.18)
j=0
which means that RI reduces the volatility of aggregate consumption growth in the risk-sensitive PIH model. Figure 5 illustrates how the combination of RI and RS affects the smoothness ratio of aggregate consumption. It is clear that incorporating RI into the risk-sensitive PIH model can reduce the volatility of aggregate consumption and thus bring the model and the data closer along this dimension. For example, when α = 5 × 10−5 , ωε = 5.6, R = 1.01, and θ = 30%, the excess smoothness ratio is about 0.44, very close to its empirical counterpart in the US data. However, λ = 0 is a strong assumption as consumers in most countries do not live in isolated islands and they interact with each other by conversations, reading the same newspapers, watching the same TV programs, and so on. We therefore need to consider the more general case in which λ ∈ (0, 1]. In this case, the volatility of aggregate consumption relative to income can be written as
µ=
βR2 − 1 θ βR2 − αω 2η
v uX u∞ 2 t Γ + λ2 j
j=0
where we use the fact that ω 2ξ = var [ξ t ] =
∞
X 1−θ (Γj − RΓj−1 )2 , 2 θ [1 − (1 − θ) R ]
1−θ ω2. θ[1−(1−θ)R2 ] ε
(3.19)
j=0
Figure 6 illustrates how the combination
of θ and λ affects the smoothness ratio of aggregate consumption when α = 5 × 10−5 , ω ε = 5.6, and R = 1.01. It is clear that the aggregation factor, λ, increases the volatility of aggregate consumption because the presence of the common noise, ξ t , offsets the smoothness of consumption caused by the gradual responses to fundamental shocks. Some combinations of (θ, λ) can still explain the observed smoothness of aggregate consumption. For example, when θ = 0.3 and λ = 0.3, the excess smoothness ratio is about 0.48, also very close to its empirical counterpart. 3.2. Robust-RI Model We now consider the connections between risk sensitivity and robustness. Hansen and Sargent (2007) demonstrate a deep connection between the choices made by an agent that is risk-sensitive and one that has a preference for robustness. An agent with a preference for robust decision rules considers a range of models and makes decisions that maximize utility given the worst possible model. Following 23
Hansen and Sargent (2007), the robust PIH problem with inattentive consumers can be written as
1 vb (b st ) = max min − (ct − c)2 + βEt ϑν 2t + b v (b st+1 ) ct νt 2
(3.20)
subject to the flow budget constraint
st+1 = R (st − ct ) + ζ t+1 + ω ζ ν t
(3.21)
and the Kalman filter equation st − ct ) + ω ζ ν t ] + θ st+1 + ξ t+1 , sbt+1 = (1 − θ) [R (b
(3.22)
where ν t distorts the mean of the innovation and ϑ > 0 controls how bad the error can be.25 After substituting (3.21) into (3.22), the Kalman filter equation can be rewritten as
sbt+1 = R (b st − ct ) + ω ζ ν t + η t+1 .
(3.23)
where η t+1 = θR (st − sbt ) + θ ζ t+1 + ξ t+1 .
The following proposition summarizes the solution to the RB-RI model.
Proposition 3. Given ϑ the value function of an agent with preference for robustness under RI is given by βR2 − 1 R βR2 − 1 2 s b − cb st − 2βR2 − ω 2ζ /ϑ t R − 1 βR2 − ω2ζ / (2ϑ) 2 2 2 R βR − 1 β βR − 1 c2 + ω 2η , 2 2 2 1 − β 2βR − ω ζ /ϑ 2 βR2 − ω 2ζ / (2ϑ) (R − 1)
vb (b st ) = −
25
(3.24)
Formally, this setup is a game between the decision-maker and a malevolent nature that chooses the distortion process ν t . ϑ ≥ 0 is a penalty parameter that restricts attention to a limited class of distortion processes; it can be mapped into an entropy condition that implies agents choose rules that are robust against processes which are close to the trusted one.
24
the consumption function is 2 / (2ϑ) (R − 1) + 1 − βR2 1 − ω ζ −1 ct = c, sbt + 2 2 βR − ω ζ / (2ϑ) βR2 − ω 2ζ / (2ϑ) (R − 1) βR2
(3.25)
the optimal worst-case distribution is given by
νt =
ω 2ζ
βR2 − 1
ω 2ζ
sbt + 2ϑ βR βR2 − ω 2 / (2ϑ) 2ϑ ζ
R R−1
βR2 − 1
c. βR βR2 − ω 2ζ / (2ϑ)
(3.26)
Proof. See Appendix 7.3. Equation (3.25) displays an important property – it is independent of θ. Thus, in the RBRI model limited channel capacity does not affect consumption and precautionary savings except through changes in the level of perceived permanent income. In this model, the preference for robustness inspires precautionary savings because agents wants to protect themselves against mistakes in specifying conditional means of the innovation to permanent income. Just like in the RS model, this kind of precautionary savings does not depend on the convexity of the marginal utility of consumption. The same property holds for (3.26); the worst-case distribution does not directly depend on the channel capacity. The value function therefore only directly depends on θ through the constant term. Individual and aggregate dynamics of consumption under robustness are very similar to those under RS, so we omit an explicit discussion.
4. Observational Equivalence In this section we detail the observational equivalence results that connect the RI, RS-RI, and RB-RI models. All three models are capable of producing the same consumption-savings decisions for some combination of parameters (α, β, ϑ, θ), rendering them fundamentally unidentified by this data alone. Our purpose is to provide intuition about why the observational equivalence result obtains in three settings – between RI and RS-RI, between RI and RB-RI, and between RS-RI and RB-RI. 4.1. Observational Equivalence Between RI and RS-RI HST (1999) show that as far as the quantity observations on consumption and investment are cone of the PIH model is observationally equivalent to the cerned, the risk-sensitive version (α > 0, β) 25
standard version (α = 0, β) of the PIH model for a unique pair of discount factors.26 The intuition is that introducing a preference for risk-sensitivity (RS) or a concern about robustness (RB) increases savings in the same way as increasing the discount factor, so that the discount factor can be changed to offset the effect of a change in RS or RB on consumption and investment.27 However, under RI, both the marginal propensity to consume (MPC) and precautionary savings are determined by the interactions of RI and RS (or RB). Since there is no prior knowledge about the mechanism of the impacts of interactions of RI and RS (or RB) on MPC and precautionary savings, the OE result proposed by HST (1999) might not hold under RI. In this section we show that a generalized observational equivalence result holds under RI. Specifically, holding all parameters constant except the pair (α, β), the RI version of the PIH model with risk-sensitive consumers (α > 0 and βR < 1) is observationally equivalent to the standard RI version of the model (α = 0 and βR = 1), extending the observational equivalence result to a broader class of models. To do so, we fix R and assume that β differs across the two settings; thus, β RS,RI is the discount factor needed in the RS-RI model to achieve OE with the RI model where βR = 1. Proposition 1. Let β RS,RI =
R − (R − 1) αω 2η 1 < . 2 R R
(4.1)
Then consumption and savings are identical in the RI and RS-RI models. Proof. See Appendix 7.4. This proposition is straightforward to prove by simply setting the coefficients in the consumption function equal and solving for the discount factor. We show in the appendix that the same outcome is obtained by setting the slope and curvature coefficients in the value functions equal. (4.1) implies that the required discount factor is decreasing as a function of α and increasing as a function of θ: (R − 1) ω 2η ∂ β RS,RI = − <0 ∂α R2 ∂ (R − 1) α ∂ω 2η β RS,RI = − > 0. ∂θ R2 ∂θ 26
HST (1999) derive the observational equivalence result by fixing all parameters, including R, except for the pair (α, β). 27 As shown in HST (1999), the two models have different implications for asset prices because continuation valuations would alter as one alters (α, β) within the observational equivalence set.
26
Holding fixed θ, an agent who is more sensitive to risk (higher α) will save more; thus, to match an agent who is not sensitive to risk that agent must be less patient. Similarly, holding fixed α an agent with a higher channel capacity (higher θ) will save less, requiring them to be more patient to match the saving of an agent who is risk sensitive but has lower channel capacity. Note that RS,RI ∂ β = 0 if α = 0, so that the discount factor would not need to be adjusted if agents are ∂θ
not risk sensitive; this result simply restates the result in Luo (2008) and arises because RI does not alter savings plan when α = 0. Figure 7 plots the locus of OE between (α, β) for various different values of θ (we fix R = 1.01). 4.2. Observational Equivalence Between RI and RB-RI We now turn to a comparison of RI and RB-RI. As above, we can derive an expression that links the discount factor in the RB-RI model to the robustness parameter ϑ in such a way that the agent would make the same choices in the RI and RB-RI models. Proposition 2. Let β RB,RI =
R − (R − 1) ω 2ζ / (2ϑ) R2
<
1 . R
(4.2)
Then consumption and savings are identical in the RI and RB-RI models. Proof. See Appendix 7.4. Here, the discount factor needed in the RB-RI model does not depend on the channel capacity parameter θ; this independence is a consequence of the previously-noted independence of consumption in this model. It is immediate that the discount factor is an increasing function of ϑ: R − 1 ω 2ζ ∂ β RB,RI = > 0. ∂ϑ R2 2ϑ2 An agent with higher ϑ is less concerned about model misspecification and therefore tends to save less (equivalently, they consider only models that are very close to the trusted one, leading to less distortion and therefore less additional saving); in order to match the behavior of an agent who does not require robust decisions the robust agent must be made more patient. If ϑ = ∞ then RB,RI ∂ = 0; again, this result matches that found in Luo (2008). ∂ϑ β 27
4.3. Observational Equivalence Between RS-RI and RB-RI We now turn the final comparisons, where we provide parameter constellations for (α, β, θ, ϑ) that jointly produce observational equivalence. These results imply that there exists a fundamental lack of identification between the class of models we are considering here (linear-quadratic-Gaussian models); it would be impossible to distinguish between any of them using only consumption-savings decisions.28 Proposition 3. Let the following expression hold: αθ 1 = . 1 − (1 − θ) R2 2ϑ
(4.3)
Then consumption and savings are identical in the RS-RI and RB-RI models. Proof. See Appendix 7.4. To interpret these expressions we fix β at some common value. An agent who is risk-sensitive (α > 0) and suffers from finite Shannon channel capacity (θ < 1) will make the same choices as an agent who is concerned with robustness (ϑ < ∞). The ”required” ϑ is given by
ϑ=
where
∂ϑ ∂θ
=
R2 −1 2αθ 2
> 0 and
∂ϑ ∂α
1 − (1 − θ) R2 , 2αθ
(4.4)
2
= − 1−(1−θ)R < 0.That is, if we compare two agents with the same 2α2 θ
α, the one with higher channel capacity will behave the same as an agent who is less concerned with model misspecification. If we compare two agents with the same θ, the one with higher risk sensitivity will behave the same as an agent who is more concerned with model misspecification. Another observational equivalence result arises if we fix (α, ϑ) for two agents who have the same θ. There exists a pair of discount factors, β RI,RS and β RI,RB , such that the two agents choose the same consumption plans. Proposition 4. Let the two discount factors satisfy the relation
β
RI,RB
1 − R2 β RI,RS ω 2ζ + 2ϑ R2 β RI,RS − αω 2η = . 2ϑR2 1 − αω 2η
28 By extension, if we interpret our model as a model of capital accumulation with a linear technology, it is impossible to distinguish the models using only consumption and investment data.
28
Then consumption and savings are identical in the RS-RI and RB-RI models.
5. Welfare Costs From RI Since information-processing constraints cannot help in individuals’ optimization – agents under RI cannot observe the state perfectly when making optimal decisions – the average welfare difference between the RI and RE economies is greater than 0. We present here the welfare cost of RI – how much utility does an agent lose if the actual consumption path he chooses under RI deviates from the first-best RE consumption path? Alternatively, we ask what an agent would pay to increase channel capacity κ to ∞ (so that the optimal choice is θ = 1). Luo (2008) shows that the welfare costs of RI are fairly small in the certainty equivalent environment; our interest here is assessing how these costs differ in the RS-RI and RB-RI models.29 To make the comparisons meaningful, we restrict our attention to combinations of preferences that imply observational equivalence – that is, do agents who look the same (but for different reasons) suffer the same welfare losses from information-processing capacity limitations? 5.1. Welfare Losses in RS-RI and RB-RI We assume for this section that (α, ϑ) are such that OE obtains between the two models (note that OE requires that α depend on θ). Within this class we can derive the lifetime utility in the RS-RI model, RS
vb
R−1 2 1 sb + cb (b st ) = − st − 2R t 2
R 2 R−1 2 β 1 log 1 − αω η , c − R−1 1−βα R
(5.1)
and in the RB-RI model,
vbRB (b st ) = −
R−1 2 1 sb + cb st − 2R t 2
R 2 β R−1 2 c + ωη , R−1 1−β R
(5.2)
where β = β RS,RI = β RB,RI . The only difference between the two functions is the constant term; this equivalence arises because the slope and curvature parameters are pinned down by the equalization 29
Our calculation is related to Tallarini (2000), who shows how risk-sensitivity increases the welfare cost of business cycles and the equity premium.
29
of the consumption decisions. We note that 1 R−1 2 R−1 2 lim − log 1 − αω η = ωη , α→0 α R R
(5.3)
so that the two expressions are equal if α = 0. We also note that ∂ ∂α Thus, we have
−
1 R−1 2 log 1 − αωη > 0. α R
R−1 2 1 R−1 2 − log 1 − αωη > ωη α R R
for α > 0. This result implies that RS households have lower utility than RB agents do, conditional on being observationally equivalent. We summarize the previous discussion as a formal proposition. Proposition 1. Given the same level of the perceived state sbt and assuming (4.3) is satisfied by (α, ϑ) for fixed (θ, β), it follows that vbRS (b st ) < vbRB (b st ).
A similar proposition holds in the absence of RI simply by replacing ω 2η with ω2ζ , so that it holds
when θ = 1. This result implies that the costs of uncertainty are larger for RS agents than RB agents whenever the parameters are such that their observable behavior is identical. Both types of agents respond by saving more than a standard RI agent would; under the OE restriction for parameters, they therefore generate the same consumption path. But RS agents are more averse to the remaining fluctuations in consumption than RB agents are, resulting in higher costs of uncertainty and lower lifetime utility. Proposition 2. Define the difference between the welfare losses of RS and RB consumers as 1 R−1 2 R−1 2 ∆ = − log 1 − αω η − ωη . α R R ∆ is decreasing in θ. Proof. By straightforward differentiation we obtain ∂∆ ∂ω 2η ∂∆ = < 0, ∂θ ∂ω 2η ∂θ 30
(5.4)
because ∂ω 2η ∂∆ R−1 R−1 = − > 0 for α > 0 and < 0. ∂ω 2η R − (R − 1) αω 2η R ∂θ
RI increases the difference between the welfare losses of RS and RB consumers who have the same consumption-savings decisions; as θ decreases, the cost of RI for the RS agents increases faster than the cost for the RB agents. Furthermore, we can use (5.1) and (5.2) to compute the marginal welfare losses due to RI at different channel capacities (θ).30
Specifically, following Barro (2007), the marginal welfare costs
(mwc) due to RI in the RS and RB models can be written as
mwc RS
=
mwc RB =
where
∂b vRS ∂θ
and
∂b vRB ∂θ
∂b v RS /∂θ
=
(∂b v RS /∂b s0 ) sb0
∂b v RB /∂θ = (∂b v RB /∂b s0 ) sb0
β R−1 − 12 1−β R−(R−1)αω 2
η
− R−1 b20 + cb s0 R s 2 ∂ω β R−1 η − 12 1−β R ∂θ , − R−1 b20 + cb s0 R s
∂ω 2η ∂θ
(5.5)
(5.6)
are evaluated for given sb0 , and the welfare costs due to RI are compared with
that from a small proportional change in the initial level of the perceived state sb0 . Therefore, the following ratio, λ, measures the relative marginal welfare losses due to RI in the two economies at various capacities (θ): λ=
mwc RS R = . RB mwc R − (R − 1) αω 2η
(5.7)
Expression (5.7) clearly shows that this ratio is different at various capacities and is determined by the interaction of channel capacity and labor income uncertainty given the interest rate R: ω 2η = θ ω 2 . It is worth emphasizing that λ can 1−(1−θ)R2 ζ R−(R−1)αω 2η < R1 . Therefore, β, as determined by R2
rewritten as
1 βR ,
where β = β RS,RI = β RB,RI =
the interaction of R, α, θ, and ω 2ε , measures the
relative marginal welfare losses of RS and RB agents at the channel capacity θ. For example, using the parameters specified in Section (3.1.2), R = 1.01, α = 5 × 10−5 , and ωε = 5.6, we have λ = 1.24 when θ = 10% and λ = 1.2 when θ = 30%. The above simple calculations simply show that although RS agents suffer more from lower channel capacity than RB agents do, the relative marginal welfare losses of RS and RB agents may not be very significant. In the next subsection, we will calculate 30
We thank the anonymous referee for suggesting that we examine the marginal welfare losses due to RI under OE.
31
the welfare losses due to deviating from the first-best full information path and will then provide more quantitative results. 5.2. Quantitative Results To compute the welfare losses of RS and RB consumers from RI due to deviating from the first-best RE path, we need the costs of deviating from the RE benchmark.
Specifically, we first assume
that at the beginning both RS and RB consumers have unlimited capacity and choose the same consumption-savings decisions, that is, the OE under RE holds (α = 1/ (2ϑ)); we then increase the degree of inattention and compute the welfare costs of RI due to deviating from the RE path. After deriving the value functions in all cases, it is straightforward to show that under the OE, α = 1/ (2ϑ), the expressions for the expected welfare losses from RI are ! 2 2 1 βR2 − 1 1 E sbt − E st E [∆v (α)] = + (5.8) 2 βR2 − αω 2η βR2 − αω 2ζ ! βR2 − 1 Rc 1 1 E [st ] − 2 E [b st ] + R−1 R β − αω 2η R2 β − αω 2ζ ! 2 2 R2 β−1 1 − αω 2 2 2 2 ζ βR −αω ζ R R β−1 c 1 1 β 1 , − 2 log + 2 2 2 2 2 2 R β − αω η 1 − β 2α R β − αω ζ 2 (R − 1) 1 − αω η R2 β−1 2 βR −αω η
where β =
R−(R−1)αω 2ζ R2
for the RS-RI model and
E [∆v (ϑ)] =
where β =
R−(R−1)ω 2ζ /(2ϑ) R2
R−1 β R−1 2 (var (b st ) − var (st )) + ω η − ω 2ζ , 2R 1 − β 2R
(5.9)
for the RB-RI model.
To do quantitative welfare analysis in the models we need to know the level of E [st ] (note that it is equal to E [b st ]). First, denote by γ the local coefficient of relative risk aversion, which equals
γ=
E [y] c − E [y]
for the utility function u (·) evaluated at mean income E [y]. To calculate the welfare losses due to RI, we set the parameters according to those estimated from post-World War U.S. time series by HST (1999) for a PIH model with two-factor endowment process. Specifically, we follow the procedure 32
used in Hansen and Sargent (2004) and use the estimated one-factor endowment process as follows
yt+1 = 0.9992yt + εt+1 ,
(5.10)
and εt+1 follows an iid process distributed as N (0, 5.6). Since the estimated persistence coefficient is difficult to distinguish from 1, for simplicity we assume that income follows a random walk. Following HST (1999) we also set the mean income level E [y] = 16 and then find the value of the bliss point c that generates reasonable relative risk aversion γ. For example, if γ is equal to 1, c = 2E [y] = 32. Furthermore, assume that the ratio of mean financial wealth to mean labor income is 5, that is, E [w] /E [y] = 5.31 Since st = wt +
1 R−1 yt ,
E [st ] =
we have 5+
1 R−1
E [yt ] .
We will use this specification when we make quantitative statements. Following Cochrane (1989), Pischke (1995), and others, we use a money metric to measure the welfare cost of deviating from the standard RS and RB solutions. Specifically, dividing the expected welfare losses E [∆v] = E [v(st ; θ = 1) − vb(b st ; θ < 1)] by the marginal utility of a dollar at time t and
converting it to dollars per quarter yields
R − 1 E [∆v (α)] ; R u′ (y) R − 1 E [∆v (ϑ)] = , R u′ (y)
$ Loss/quarter:
∆RS =
(5.11)
$ Loss/quarter:
∆RB
(5.12)
where ∆v (α) and ∆v (ϑ) are defined in (5.8) and (5.9), respectively. Table 1 reports welfare costs for several values of α and the optimal weight on observations θ. It is clear from Table 1 that the welfare losses due to RI are trivial. For example, for α = 5 × 10−4 and θ = 10%, (that is, 10% of the uncertainty is removed upon the receipt of a new signal), the loss only amounts to 76 cents per quarter. This result is similar to the findings by Pischke (1995) and Luo (2008).32 Table 1 also shows that for plausible assumptions about the coefficient for local relative risk aversion (γ) and the ratio of mean 31 This number varies largely for different individuals, from 2 to 20. 5 is the average wealth/income ratio in the Survey of Consumer Finances 2001. 32 Pischke (1995) calculates utility losses in the no-aggregate-information model and finds that in most cases the utility losses due to no information about aggregate shocks are less than $1 per quarter.
33
financial wealth to mean income (E [w] /E [y]), the welfare losses due to RI are not significant even if agents are risk-sensitive.33 This thus provides some evidence that it is reasonable for agents to devote low channel capacity to observing and processing information because the welfare improvement from increasing capacity is trivial. In other words, although consumers can devote much more capacity to processing economic information and then improve their optimal consumption decisions, it is rational for them not to do so because the welfare improvement is tiny. Our main purpose in this section is to demonstrate that the welfare losses from RI can in fact be considerably larger for RS agents. To this end, Table 3 reports the ratio of the welfare losses of RS to RB, ∆RS /∆RB . We set θ = 1 and α = 1/ (2ϑ) to reflect the OE between robustness and risk-sensitivity in the RE context and then consider lower values of θ. The costs are uniformly higher for the RS model, and can in fact be much larger (the rightmost column has ratios that approach 8). Similarly, Table 4 reports the ratio of the RI-induced welfare losses in the RS-RI and basic RI models for different values of θ given observationally-equivalent consumption-savings decisions. Note that when calculating the welfare losses in the RS model, the value of β is determined by (4.1) given the values of α and θ, while the value of β used in the basic RI model is just 1/R. Risksensitive consumers also suffer much more from finite information-processing constraints than do the consumers who are simply more patient. It is worth noting that as (4.3), shows, changing channel capacity while holding RS and RB fixed would break the OE; consequently, the welfare consequences reported above are partly affected by the different consumption-savings rules. What is the significance of these calculations? The fact that RS agents suffer much more from limited information processing capacity suggests that a careful experiment might be able to distinguish the two types of agents. If a researcher were to ask individuals ’What would you pay to receive more/better information about the state of the economy?’ RS and RB agents would provide significantly different answers. Given the answers to this question, along with consumption-savings data, we may be able to determine whether agents are risk-sensitive or concerned about model misspecification, provided they in fact have limited information-processing capacity. 33
Changing the value of E [w] /E [y] only has minor effects on quantitative results and does not affect the main results of the RS-RI model. See Table 2.
34
5.3. Implications for Countercyclical Policies Finally, it is worth noting that we can also distinguish RS and RB by examining different welfare gains of stabilization policy aimed at reducing aggregate fluctuations for RS-RI consumers and RBRI consumers. The traditional way to calculate welfare costs of business cycles proposed by Lucas (1987) is to offer risk averse consumers two possible consumption streams, one of which is constant and the other has the same mean but fluctuates. Consequently, risk averse consumers would always prefer the constant consumption stream and thus require some consumption compensation to accept the fluctuating path. As shown above, RI introduces additional uncertainty into RS and RB models and thus amplifies the impacts of RS and RB on total uncertainty faced by the consumers as well as precautionary savings. Since counter-cyclical policies (stabilization policies) which reduce aggregate fluctuations affect our model economy by reducing ω 2ζ , RS-RI consumers gain more welfare from countercyclical policies than do RB-RI consumers.34
Therefore, we can also identify RS and RB
consumers by asking them that how much they would pay to move to a world in which there is no aggregate fluctuations.
6. Conclusion This paper provides a characterization of the consumption-savings behavior of a single agent who is both risk-sensitive and limited in their capacity to process signals. The key component to our model is the presence of precautionary savings, permitting the limited processing capacity to affect the average level of consumption. We show that there still exists an observational equivalence between models with and without risk-sensitivity (and/or robustness) even when agents face informationprocessing constraints, extending the results of HST (1999) to a broader class of models. However, within the observationally-equivalent class of models the welfare costs of rational inattention are not constant – models with higher discount factors and correspondingly lower risk-sensitivity generate larger welfare costs of RI than observationally-equivalent ones with lower discount factors and higher risk-sensitivity. We also show a connection between robustness, risk-sensitivity, and the discount factor, in the sense that any combination of the three can plausibly generate the same consumption34 Note that ω 2ζ includes both idiosyncratic and aggregate components. Countercyclical policies can reduce aggregate fluctuations by either reducing individual income risk directly or reducing the correlation across individuals in their income risk.
35
savings decisions in the RI context. However, as in the case with RS and RI, the welfare costs of RI are not equivalent across these observably-equivalent specifications. While our model does permit precautionary savings, it is not as general as we would prefer. As we have noted above, solving fully-nonlinear versions of the RI model is extremely difficult given the shape of the optimal distribution of consumption, meaning that only models with short horizons or very small state spaces can be computed.35 We suspect that methods to deal with this problem can be developed, particularly for simple one-agent problems like the one studied here. A more difficult problem with RI models is noted in Sims (2005) – a general equilibrium setting will involve nonstandard features including inventories, retailing, and probably search. Implementation of RI solutions is also problematic – under general settings consumption is not a deterministic function, meaning that the theory does not make clear predictions about what any individual’s consumption will be, only what will prevail in a dataset of sufficiently large size. These difficulties will need to be overcome in order to make rational inattention a useful component of economic models. Because our model is not necessarily general, we think it advisable to comment on whether the observational equivalence results are likely to hold in more general environments. The precautionary savings induced by a concern about robustness (or enhanced risk aversion) differs from the usual precautionary savings motive that emerges when labor income uncertainty interacts with the convexity of the marginal utility of consumption. This type of precautionary savings emerges because consumers need to save more to protect themselves against model misspecification and occurs even in models with quadratic utilities. Most existing robustness models assume that the objective functions are quadratic and the state transition equation is linear, consequently, worst case distributions are Gaussian. However, if the objective functions are not quadratic or the transition equations are not linear, worst case distributions are generally non-Gaussian. As discussed in Hansen and Sargent (2007), the most difficult part in solving such non-LQ models is computational: representing the worst-case distribution parsimoniously enough that the model is tractable. Some progress in establishing a connection between risk-sensitivity and robustness has been made, though.
In a continuous-time non-LQ setting, Maenhout (2004) shows that robustness is
observationally equivalent to Duffie-Epstein-Zin recursive preferences (which contain risk-sensitive 35
Sims (2006), Lewis (2006), Batchuluun, Luo, and Young (2008), and Tutino (2008) present some nonlinear solutions to RI models. These models are generally very small, either in terms of the number of states or the number of periods.
36
preferences as a strict subset): an investor with a preference for robustness measured by α and CRRA utility function
Ct1−γ −1 1−γ
is observationally equivalent to a Duffie-Epstein-Zin investor with EIS
1 γ
and
coefficient of relative risk aversion γ + α, in the sense that they generate the same consumption and portfolio choice rules.36 Anderson, Hansen, and Sargent (1998) extend the analysis of Hansen, Sargent, and Tallarini (1999) to non-LQ continuous-time economies that solve optimal resource allocation problems and also demonstrate that concerns about robustness can imitate risk sensitivity in the sense that they lead to the same resource allocation. However, to the best of our knowledge, it seems impossible to theoretically establish the OE in the discrete-time RE non-LQ setting, as most of them have to be solved numerically. For example, we even do not have an explicit solution for a simplest discrete-time permanent income model with the CRRA utility
Ct1−γ −1 1−γ
and budget
constraint Wt+1 = R (Wt − Ct ) + Yt+1 , where Ct , Wt , and Yt are consumption, wealth, and labor income, respectively. Introducing RB or RS into this model would complicate the problem, meaning that the explicit solutions essential to establishing observational equivalence would not exist. Under RI, it is even more difficult to establish the OE between RS and RB outside the LQ framework.
RI means that imperfect information emerges endogenously and greatly complicates
RB or RS models. The main reason is that the effective state in such models is not traditional state variable, but is the so-called informations state that is defined as the distribution of the true state conditional on processed information. In other words, it expands the state space to the space of distributions on the true state (in our paper, st ), creating a severe “curse of dimensionality” problem. Within the LQ-Gaussian framework the conditional distributions are Gaussian, so that the first two moments, the conditional mean Et [st ] and the conditional covariance matrix vart (st ), are sufficient to characterize the information state. However, it is extremely difficult to solve non-LQ versions of the RI model, as the state of the world is a distribution and this distribution is not wellbehaved except in rare circumstances (Sims 2005 finds one case where the posterior is distributed as a F random variate, but this result only holds for a uniform prior and thus would not extend to horizons longer than two periods); consequently, only models with short horizons or very small state spaces can be computed, and those only to limited accuracy. It seems unlikely that observational equivalence can be proven. Some current work is focused on establishing an approximate result – 36 Backus, Routledge, and Zin (2004) also comment that the connection between the two models would hold outside the confines of the LQ-Gaussian model, but do not provide any formal proof.
37
how the introduction of RI alters our estimates of preference parameters. Batchuluun (2009) and Batchuluun, Luo, and Young (2008) contain preliminary results on the connections between risk aversion, intertemporal substitution in consumption and labor supply, and channel capacity. We conclude with some comments on the consequences of our results. Risk-sensitivity and concern for robustness are aspects of preferences; government policy should not be used to ”counteract” their effects on consumption. But rational inattention is something different – it is an additional constraint on household actions that reduces welfare. Thus, the effects of rational inattention should be attenuated via policy whenever such actions are cost-effective and feasible. Our results suggest that it is important to dig deeper into how rational inattentive agents differ from other agents, for the optimal design of policies – indeed, even whether government policy is desirable or not – will depend on why agents choose the way they do. Finally, there may be substantive differences across the various models that only manifest themselves in asset markets.
In this paper we study a set of permanent income models in which the
interest rate is constant. In a more realistic setting in which there are multiple assets and agents have to make portfolio choices, we can examine the distinct implications of the risk-sensitivity preference (a special case of the recursive utility theory) and rational inattention for long-run consumption risk – measured by the covariance of the asset return and consumption growth over the period of the return and many following periods.
Parker and Julliard (2005) and Hansen, Heaton, and Li
(2008) find strong empirical support for long-run consumption risk. As shown in Hansen, Heaton, and Li (2008), recursive preferences provide a preference-based role for long-run consumption risk, as temporal dependence in consumption growth alters risk premia.
In contrast, RI due to finite
information-processing capacity affects the intertemporal composition of risk, as the RI model predicts that consumption reacts to the innovation to the equity return gradually and with delay (see Luo 2009). In the future it would be promising to examine how the two hypotheses lead to different implications for optimal portfolio choice and the equilibrium risk premium.
38
7. Appendix 7.1. Solving the RS model To solve the Bellman equation (2.14), we conjecture that v (st ) = −As2t − Bst − C,
(7.1)
where A, B, and C are undetermined coefficients. We can then evaluate Et [exp (−αv (st+1 ))] to obtain
Et [exp (−αv (st+1 ))] = Et exp αAs2t+1 + αBst+1 + αC h i = Et exp αAR2 (st − ct )2 + αBR (st − ct ) + [2αAR (st − ct ) + αB] ζ t+1 + αAζ 2t+1 + αC b2 −1/2 exp a + = (1 − 2c) , 2 (1 − 2c) where a = αAR2 (st − ct )2 + αBR (st − ct ) + αC,
(7.2)
b = [2αAR (st − ct ) + αB] ωζ ,
(7.3)
c = αAω 2ζ .
(7.4)
Thus, the distorted expectations operator can be written as 1 Rt [v (st+1 )] = − α
1 b2 − log (1 − 2c) + a + 2 2 (1 − 2c)
2ω2 2 αB 1 AR BR ζ . = log 1 − 2αAω 2ζ − (st − ct )2 − (st − ct ) − C + 2α 2 1 − 2αAω 2ζ 1 − 2αAω 2ζ 2 1 − 2αAω ζ
Maximizing the RHS of (2.14) with respect to ct yields the first-order condition − (ct − c) +
2βAR2 BβR (st − ct ) + = 0, 2 1 − 2αAω ζ 1 − 2αAω 2ζ
39
(7.5)
which means that ct =
2AβR2 1 − 2αAω 2ζ + 2AβR2
st +
c 1 − 2αAω 2ζ + BβR 1 − 2αAω 2ζ + 2AβR2
.
(7.6)
Substituting (7.6) and (7.1) into (2.14) gives !2 2 2c 1 2AβR BβR − 2AβR − As2t − Bst − C = − st + + 2 1 − 2αAω 2ζ + 2AβR2 1 − 2αAω 2ζ + 2AβR2 2 1−2αAω 2ζ c(1−2αAω 2ζ )+BβR 1 AR2 2 2α log 1 − 2αAω ζ − 1−2αAω2ζ 1−2αAω2ζ +2AβR2 st − 1−2αAω2ζ +2AβR2 . β 1−2αAω 2ζ c(1−2αAω 2ζ )+BβR αB 2 ω 2ζ BR − 1−2αAω2 1−2αAω2 +2AβR2 st − 1−2αAω2 +2AβR2 − C + 2 1−2αAω 2 ( ζ ζ ζ ζ)
Collecting and matching terms, the constant coefficients turn out to be
A=
βR2 − 1 , 2βR2 − 2αω 2ζ
(7.7)
R 1 − βR2 B= c, βR2 − αω 2ζ (R − 1) R2 βR2 − 1 1 β 1 c2 + C= log 1 − 2αAω 2ζ , 2 (R − 1)2 βR2 − αω 2 1 − β 2α ζ
(7.8)
(7.9)
Substituting (7.7) and (7.8) into (7.6) yields the consumption function (2.17) in the text. 7.2. Solving the RI-RS PIH model To obtain the solution to the RI-RS model we simply need to replace ω 2ζ with ω 2η (we omit the steps): A=
βR2 − 1 2βR2 − 2αω 2η
R 1 − βR2 B= c βR2 − αω 2η (R − 1) R2 βR2 − 1 1 β 1 2 2 C= c − log 1 − 2αAω η 2 (R − 1)2 βR2 − αω 2η 1 − β 2α
40
and the consumption function is 1 − αω 2η (R − 1) + 1 − βR2 βR2 − 1 ct = sbt + c βR2 − αω 2η βR2 − αω 2η (R − 1)
(7.10)
7.3. Solving the Robust RI Model (Not for Publication)
As before, we conjecture that vb (b st ) = −Ab s2t − Bb st − C, where A, B, and C are undetermined coefficients. Substituting this guessed value function, (3.21), and (3.22) into the Bellman equation gives 1 −Ab s2t − Bb st − C = max min − (c − ct )2 + βEt ϑν 2t − Ab s2t+1 − Bb st+1 − C . ct νt 2
(7.11)
We can do the min and max operations in any order, so we choose to do the minimization first. The first-order condition for ν t is
2ϑν t − 2AEt [ω ζ ν t + R (b st − ct )] ωζ − Bω ζ = 0, which means that νt =
B + 2AR (b st − ct ) ωζ . 2 ϑ − Aω 2ζ
(7.12)
Substituting (7.12) back the Bellman equation (3.20) gives
2 1 B + 2AR (b st − ct ) 2 2 2 −Ab st − Bb st − C = max − (c − ct ) + βEt ϑ ω ζ − Ab st+1 − Bb st+1 − C , ct 2 2 ϑ − Aω 2 ζ
(7.13)
where
sbt+1 = (1 − θ) [R (b st − ct ) + ω ζ ν t ] + θ R (st − ct ) + ωζ ν t + ζ t+1 + ξ t+1 = R (b st − ct ) + ω ζ ν t + η t+1
The first-order condition for ct is (c − ct ) + 2βA R +
ARω 2ζ ϑ − Aω 2ζ
!
[R (b st − ct ) + ωζ ν t ] + βB
41
R+
ARω 2ζ ϑ − Aω 2ζ
!
− 2βϑ
ARω ζ ν t = 0. ϑ − Aω 2ζ
Using the solution for ν t the solution for consumption is
ct =
2AβR2 1 − Aω 2ζ /ϑ + 2AβR2
sbt +
c 1 − Aω 2ζ /ϑ + BβR 1 − Aω 2ζ /ϑ + 2AβR2
.
(7.14)
Substituting the above expressions into the Bellman equation gives st − C − Ab s2t − Bb 2 2 2AβR2 −2AβR2 c+BβR 1 1 AR B−2ARc − 2 1−Aω 2 /ϑ+2AβR2 sbt + 1−Aω2 /ϑ+2AβR2 + βϑ 2 2AβϑR2 −Aω2 +ϑ + ϑ−Aω2 +2AβR2 ϑ sbt ω 2ζ ζ ζ ζ 2ζ 2 +2cϑR−Bω 2 2BβϑR ζ Rϑ 1 = −A 2AβϑR2 −Aω2ζ +ϑ sbt − 2 2AβϑR2 −Aω2ζ +ϑ + η t+1 +βEt 2 2 2BβϑR +2cϑR−Bω ζ Rϑ 1 −B 2AβϑR2 −Aω2 +ϑ sbt − 2 2AβϑR2 −Aω2 +ϑ + η t+1 − C ζ
ζ
(7.15)
Collecting and matching terms, the constant coefficients turn out to be
A=
βR2 − 1 , 2βR2 − ω 2ζ /ϑ
(7.16)
R 1 − βR2 c, R − 1 βR2 − ω 2ζ / (2ϑ) R2 βR2 − 1 1 β βR2 − 1 C= c2 + ω2 . 2 βR2 − ω 2 / (2ϑ) (R − 1)2 1 − β 2βR2 − ω 2ζ /ϑ η ζ
B=
(7.17) (7.18)
Substituting (7.16) and (7.17) into (7.14) yields the consumption function (3.25) in the text. 7.4. Observational Equivalence Between RI-RS and RI-RB (Not for Publication) Comparing (3.10) with (3.25), we can derive the OE between RS and RB under RI: αω 2η = ω 2ζ / (2ϑ) ,
(7.19)
which implies that α
θ 1 = . 1 − (1 − θ) R2 2ϑ
Note that when βR = 1, the standard RI PIH model implies that ct =
42
(7.20) R−1 bt . R s
Hence, under
RS-RI (α > 0), we can construct the following OE between α and β βR2 − 1 R−1 = , 2 2 βR − αω η R which implies that β RI,RS =
R − (R − 1) αω 2η 1 < ; 2 R R
(7.21)
Similarly, under RS-RI (ϑ > 0), we can construct the following OE between β and ϑ βR2 − 1 R−1 = , 2 2 R βR − ω ζ / (2ϑ) which implies that β RI,RB =
R − (R − 1) ω 2ζ / (2ϑ) R2
<
1 . R
(7.22)
Given the OE under RI, the value function under RB-RI can be written as RB
vb
R−1 2 (b st ) = − sb + cb st − 2R t
1 R 2 β R−1 2 c + ω , 2R−1 1 − β 2R η
and the value function under RS-RI can be written as vbRS (b st ) = −
R−1 2 sb + cb st − 2R t
1 R 2 β 1 c − log 1 − 2αAω 2η . 2R−1 1 − β 2α
Therefore, the only difference between the two value functions is the constant term. Note that R−1 2 1 R−1 2 lim − log 1 − αω η = ωη . α→0 α R R and 1 ∂ − 2α log 1 − ∂α
R−1 2 R αω η
! − (R − 1) αω2η R−1 2 − log 1 − αωη R − (R − 1) αω 2η R ! − (R − 1) αω2η (R − 1) αω 2η 1 ≃− 2 + 2α R − (R − 1) αω 2η R 2 (R − 1) αω 2η 1 > 0; = 2α2 R R − (R − 1) αω 2η 1 =− 2 2α
43
the second line follows whenever
R−1 2 R αω η
is a small number. Hence,
1 R−1 2 R−1 2 − log 1 − αω η > ω for α > 0, 2α R 2R η that is, RS consumers suffer larger welfare losses from RI than do RB consumers.
44
(7.23)
References [1] Anderson, Evan W., Lars Peter Hansen, and Thomas J. Sargent (1998), “Risk and Robustness in General Equilibrium,” mimeo, University of Chicago. [2] Backus, David K., Bryan R. Routledge, and Stanley E. Zin (2004), “Exotic Preferences for Macroeconomists,” NBER Macroeconomics Annual 2004, pp. 319-414. [3] Barillas, Francisco, Lars Peter Hansen, and Thomas J. Sargent (2008), “Doubts or Variability?” mimeo, New York University. [4] Barro, Robert (2007), “On the Welfare Costs of Consumption Uncertainty,” mimeo, Harvard University. [5] Batchuluun, Altantsetseg (2009), “Labor Supply with Information-Processing Constraints,” mimeo, University of Virginia. [6] Batchuluun, Altantsetseg, Yulei Luo, and Eric R. Young (2008), “Portfolio Choice with Information-Processing Limits,” mimeo, University of Virginia and Hong Kong University. [7] Carroll, Christopher, and Andrew Samwick (1997), “The Nature of Precautionary Wealth,” Journal of Monetary Economics 40(1), 41–71. [8] Cochrane, John H. (1989), “The Sensitivity of Tests of the Intertemporal Allocation of Consumption to Near-Rational Alternatives,” American Economic Review 79(3), pp. 319-37. [9] Cover, Thomas M. and Joy A. Thomas (1991), Elements of Information Theory, John Wiley and Sons. [10] Duffie, Darrell and Larry G. Epstein (1992), “Stochastic Differential Utility,” Econometrica 60(2), pp. 353–94. [11] Eeckhoudt, Louis and Harris Schlesinger (2008), “Changes in Risk and the Demand for Saving,” Journal of Monetary Economics 55(7), pp. 1329-36. [12] Epstein, Larry G. and Stanley E. Zin (1989), “Substitution, Risk Aversion, and the Temporal Behavior of Consumption and Asset Returns: A Theoretical Framework,” Econometrica 57(4), pp. 937-69. 45
[13] Flavin, Marjorie A. (1981), “The Adjustment of Consumption to Changing Expectations About Future Income,” Journal of Political Economy 89(5), pp. 974-1009. [14] Friedman, Milton (1957), A Theory of the Consumption Function, National Bureau of Economic Research. [15] Guiso, Luigi, Tullio Jappelli and Daniele Terlizzese (1992), “Earning Uncertainty and Precautionary Saving,” Journal of Monetary Economics 30(2), pp. 307-38. [16] Hall, Robert E. (1978), “Stochastic Implications of the Life Cycle-Permanent Income Hypothesis: Theory and Evidence,” Journal of Political Economy 86(6), pp. 971-87. [17] Hansen, Lars Peter, John C. Heaton, and Nan Li (2008), ”Consumption Strikes Back? Measuring Long Run Risk,” Journal of Political Economy 116(2), pp. 260-302. [18] Hansen, Lars Peter and Thomas J. Sargent (1995), “Discounted Linear Exponential Quadratic Gaussian Control,” IEEE Transactions on Automatic Control 40, pp. 968-71. [19] Hansen, Lars Peter and Thomas J. Sargent (2004), “’Certainty Equivalence’ and ’Model Uncertainty,”’ working paper, University of Chicago and New York University. [20] Hansen, Lars Peter and Thomas J. Sargent (2007), Robustness, Princeton University Press. [21] Hansen, Lars Peter, Thomas J. Sargent, and Thomas D. Tallarini, Jr. (1999), “Robust Permanent Income and Pricing,” Review of Economic Studies 66(4), pp. 873-907. [22] Hansen, Lars Peter, Thomas J. Sargent, and Neng Wang (2002) “Robust Permanent Income and Pricing with Filtering,” Macroeconomic Dynamics 6(1), pp. 40-84. [23] Jacobson, David H. (1973), “Optimal Stochastic Linear Systems with Exponential Performance Criteria and Their Relation to Deterministic Differential Games,” IEEE Transactions on Automatic Control 18, pp. 124-31. [24] Kasa, Kenneth (2006), “Robustness and Information Processing,” Review of Economic Dynamics 9(1), pp. 1-33. [25] Lewis, Kurt F. (2006), “The Life-Cycle Effects of Information-Processing Constraints,” mimeo, University of Iowa. 46
[26] Lucas, Robert E. (1987), Models of Business Cycles, New York: Basil Blackwell. [27] Luo, Yulei (2008), “Consumption Dynamics Under Information Processing Constraints,” Review of Economic Dynamics 11(2), pp. 366-85. [28] Luo, Yulei (2009), ”Rational Inattention, Long-run Consumption Risk, and Portfolio Choice,” mimeo, Hong Kong University. [29] Maenhout, Pascal (2004), “Robust Portfolio Rules and Asset Pricing,” Review of Financial Studies 17(4), pp. 951-83. [30] Parker, Jonathan A. and Christian Julliard (2005), ”Consumption Risk and the Cross Section of Expected Returns,” Journal of Political Economy 113(1), pp. 185-222. [31] Pischke, Jorn-Steffen (1995), “Individual Income, Incomplete Information, and Aggregate Consumption,” Econometrica 63(4), pp. 805-40. [32] Reis, Ricardo (2006), “Inattentive Consumers,” Journal of Monetary Economics 53(8), pp. 1761-800. [33] Shannon, Claude (1948), “A Mathematical Theory of Communication,” The Bell System Technical Journal 27, pp. 623-56. [34] Sims, Christopher A. (1998), “Stickiness,” Carnegie-Rochester Conference Series on Public Policy 49, pp. 317-56. [35] Sims, Christopher A. (2003), “Implications of Rational Inattention,” Journal of Monetary Economics 50(3), pp. 665-90. [36] Sims, Christopher A. (2005), “Rational Inattention: A Research Agenda,” Technical Report, Princeton University. [37] Sims, Christopher A. (2006), “Rational Inattention: Beyond the Linear-Quadratic Case,” American Economic Review 96(2), pp. 158-63. [38] Tallarini, Thomas D., Jr. (2000), “Risk-Sensitive Real Business Cycles,” Journal of Monetary Economics 45(3), pp. 507-32.
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[39] Tutino, Antonella (2008), “The Rigidity of Choice:
Life Cycle Savings with Information-
Processing Limits,” mimeo, Princeton University. [40] van der Ploeg, Frederick (1993), “A Closed-Form Solution for a Model of Precautionary Saving,” Review of Economic Studies 60(2), pp. 385-95. [41] Wang, Neng (2004), “Precautionary Saving and Partially Observed Income,” Journal of Monetary Economics 51(8), pp. 1645-81. [42] Whittle, Peter (1981), “Risk-Sensitive Linear/Quadratic/Gaussian Control,” Advances in Applied Probability 13, pp. 764-77. [43] Whittle, Peter (1990), Risk-Sensitive Optimal Control, John Wiley and Sons.
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Figure 1: Response of Consumption to Income Shock
1.3 RE RS (α=0.0005) RI (θ=50%) RS+RI (θ=50%, α=0.0005)
1.2 1.1 1 0.9 0.8 0.7 0.6
0
5
10 Period
49
15
20
Figure 2: MPC out of Financial Wealth w and Human Wealth
y R−1
0.014
0.012
α=0 α=0.0001 α=0.0002 α=0.0005
0.01
0.008
0.006
0.004
0.002
0 0.1
0.2
0.3
0.4
0.5
θ
0.6
50
0.7
0.8
0.9
1
Figure 3: Effect of (α, θ) on Precautionary Savings
18 16 14 12 α=0 α=0.0001 α=0.0002 α=0.0005
10 8 6 4 2 0 0.1
0.2
0.3
0.4
0.5
θ
0.6
51
0.7
0.8
0.9
1
Figure 4: Short-Run Response of Consumption
1.4
1.2
α=0 α=0.0001 α=0.0002 α=0.0005
1
0.8
0.6
0.4
0.2
0 0.1
0.2
0.3
0.4
0.5
θ
0.6
52
0.7
0.8
0.9
1
Figure 5: Effect of (α, θ) on Smoothness of Aggregate Consumption
1.2 α=0 α=0.0001 α=0.0005
1.1 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1
0.2
0.3
0.4
0.5
θ
0.6
53
0.7
0.8
0.9
1
Figure 6: Effect of (θ, λ) on Smoothness of Aggregate Consumption
−3
1 0.9 0.8 0.7
x 10
θ=30% θ=50% θ=100%
α
0.6 0.5 0.4 0.3 0.2 0.1 0
0.991 0.992 0.993 0.994 0.995 0.996 0.997 0.998 0.999 β
54
1
Figure 7: Observational Equivalence Between α and β
50 45
µ=0 µ=0.5 µ=0.9
the relative volatility, rv
40 35 30 25 20 15 10 0.1
0.2
0.3
0.4
0.5 0.6 0.7 channel capacity, θ
55
0.8
0.9
1
Table 1: Welfare losses under RS-RI, ∆RS ($/quarter) α = 10−4
α = 5 · 10−4
α = 10−3
θ = 10%
0.1389
0.7566
3.1455
θ = 40%
0.0193
0.0984
0.3263
θ = 70%
0.0054
0.0272
0.0885
θ = 90% 0.0014 0.0070 0.0226 Note. The parameters are set as follows: c = 32, E [y] = 16, E [w] /E [y] = 5, R = 1.01, γ = 1, and ω ε = 0.2559. Table 2: Welfare losses under RS-RI, ∆RS ($/quarter) α = 10−4
α = 5 · 10−4
α = 10−3
θ = 10%
0.1289
0.6853
2.8378
θ = 40%
0.0179
0.0891
0.2944
θ = 70%
0.0050
0.0247
0.0798
θ = 90% 0.0013 0.0063 0.0204 Note. The parameters are set as follows: c = 32, E [y] = 16, E [w] /E [y] = 10, R = 1.01, γ = 1, and ω ε = 0.2559.
56
Table 3: Comparison of RI-induced welfare losses the RS-RI and RB-RI models ∆RS /∆RB (α, ϑ) = 10−5 , 5 · 104
5 · 10−5 , 104 4.7313
10−4 , 5 · 103 7.9944
5 · 10−4 , 103
θ = 10%
2.5040
69.8826
θ = 40%
2.5013
4.7050
7.9028
64.5429
θ = 70%
2.5012
4.7023
7.8928
63.9736
θ = 90% 2.5009 4.7010 7.8891 63.7887 Note. The parameters are set as follows: c = 32, E [y] = 16, E [w] /E [y] = 5, R = 1.01, γ = 1, and ω ε = 0.2559. Table 4: Comparison of RI-induced welfare losses in the RS-RI and basic RI models ∆RS /∆β
θ = 10%
(α, β) = 10−5 , 0.9901 2.4757
5 · 10−5 , 0.9901 4.4690
10−4 , 0.9901 7.1318
38.8461
θ = 40%
2.4730
4.4442
7.0501
35.8781
θ = 70%
2.4728
4.4416
7.0411
35.5598
5 · 10−4 , 103
θ = 90% 2.4726 4.4404 7.0379 35.4621 Note. The parameters are set as follows: c = 32, E [y] = 16, E [w] /E [y] = 5, R = 1.01, γ = 1, and ω ε = 0.2559.
57
